Simplicial isometric embeddings of indefinite metric polyhedra

TL;DR

This paper extends isometric embedding results to indefinite metric polyhedra, showing they can be embedded into Minkowski space with bounded dimension, and explores relaxations that reduce embedding dimension requirements.

Contribution

It introduces two definitions of indefinite metric polyhedra, proves embeddings into Minkowski space under these definitions, and analyzes dimension bounds and relaxations.

Findings

Every indefinite metric polyhedron with bounded degree admits a simplicial isometric embedding into Minkowski space.

The two definitions of indefinite metric polyhedra are shown to be equivalent.

Relaxing to piecewise linear embeddings reduces the necessary dimension bounds.

Abstract

In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called "indefinite metric polyhedra". Two definitions for an indefinite metric polyhedron are given, an intuitive definition and a more useful definition. The more useful definition is used to show that every indefinite metric polyhedron (with the maximal degree of every vertex bounded above) admits a simplicial isometric embedding into Minkowski space of an appropriate signature. This result is then used to show that the two definitions coincide. A simple example is given to show that the dimension bounds in the compact case are essentially sharp. Finally it is shown that if we relax our embedding conditions to be pl instead of simplicial, then the necessary dimension bounds can be reduced to a surprisingly low signature.